On the Theory of Beams Resting on a Yielding Foundation.

Introduction.-The purpose of this note is to advance the theory of the following problem: An elastic beam of infinite length, loaded inside a finite region, rests on a yielding foundation. It is asked how much the results of this theory are influenced by alterations of the assumptions concerning the physical nature of the foundation. As it is known, in engineering the assumption is made that at every point the deflection is proportional to the foundation pressure at this point and independent of the pressure at points distant from the considered point. This assumption, which is justified for a beam resting on water, has the advantage of leading to the solution of a linear differential equation. In reality the deflection at one point depends almost always on the pressure distribution along the whole beam and the question leads, as it will be seen, to an integrodifferential equation. On this problem a very interesting paper has recently been published by M. A. Biot.' There the foundation is represented by a semi-infinite elastic body. Here it shall be shown that the formal results of that paper can be obtained in a different, more general way; more general in that it is shown to be possible to solve the problem explicitly for every kind of foundation whether elastic or having properties more difficult to describe, provided the deflection of the surface of the foundation due to a concentrated loading is known. It will also be shown that under this assumption the problem of buckling of a beam of infinite length, resting on a yieldiug foundation and on an infinity of equidistant supports, can be solved completely. So far as known a treatment of this problem has been given only for the beam resting on water.2 Statement of the Problem.-The differential equation for the deflection w of an elastic beam of constant stiffness EbJ is given by